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How we define the Lie symmetry of a stochastic differential equation?

In "On Lie-point symmetries for Ito stochastic differential equations", they explain some of the difficulties of doing that and some approaches eg. by Kozlov in "Symmetry of systems of ...
Thomas Kojar's user avatar
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1 vote

Ito's lemma and Stratonovich calculus

Based on Kurt's answer and comment, the mistake I made in Eq. (2) in the post can be resolved in an alternative way below. From Ito's lemma, we have $$ df = \left( \mu f' + \frac{1}{2} \sigma^2 f'' \...
Fred's user avatar
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4 votes
Accepted

Ito's lemma and Stratonovich calculus

In this answer I show that for any process $Y_t$ whose stochastic integrals are defined we have $$ \tag{1} \underbrace{\int_0^tY_s\circ\,dW_s}_{\text{Stratonovich}}=\underbrace{\int_0^tY_s\,dW_s}_{\...
Kurt G.'s user avatar
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2 votes

Can an SDE hit a point where its covariance is not positive definite?

I can think of an example that, as advertised in the comments, fails to be locally Lipschitz at $a=0$. Consider the following dynamics for a Cox-Ingersoll-Ross (CIR) process: $$d v_t = \sqrt{\max (v_t,...
Jose Avilez's user avatar
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1 vote

Why Do we need a right-continuous filtration with $\mathcal{F}_0$ containing all P-null sets?

In Karatzas-Shreve 2.7 "Brownian filtrations" they go over the motivation for augmenting by null-sets and the right continuity. See also Properties of Feller Processes. Here are some reasons:...
Thomas Kojar's user avatar
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1 vote
Accepted

Find the solution of SDE $dY_t=rdt+\alpha Y_tdB_t$

Consider $X_t=\alpha X_tdW_t$, $X_0=1$. Then, $X_t=e^{-(\alpha^2/2)t+\alpha W_t}$. The hint then suggests to consider $Z_t=X_t^{-1}Y_t$. Note that by Ito $$d(X_t^{-1})=-X_t^{-2}dX_t+\alpha^2X_t^{-3}...
Snoop's user avatar
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1 vote

Distribution of solution to linear SDE

We follow Integrating factor and homogeneous equation for SDEs., but write the details for this specific case \begin{align*} \mathrm{d}X_t = a(t)X_t \mathrm{d}t + d(t)\mathrm{d}B_t. \end{align*} We ...
Thomas Kojar's user avatar
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0 votes

Given a geometric Brownian motion, obtain a recurrence relation for moment $\mathbb{E}X_k^q$ and deduce $\mathbb{E}_k^q=\alpha^k(ah^{1/2},q)$

Taking the $q$th power and expectation of the Euler-Maruyama scheme for this equation gives $$\mathbb{E}X_{k+1}^q=\mathbb{E}\left(1+a\Delta W_k\right)^q\mathbb{E}X_k^q$$ Using the initial condition, ...
Thomas Kojar's user avatar
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0 votes

How to solve system of stochastic differential equations?

As mentioned in the comments the SDE $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ is decoupled and so one can solve it $$N_{1,t}=N_{1,0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{1,t}\...
Thomas Kojar's user avatar
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2 votes
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Question about Ito's lemma

This holds as long as $X$ is right-continuous at $0$ (assuming you meant $h$ instead of $2h$ in the denominator). I'll just show it for $d=1$ and $Y_0 = 0$, but the general case works exactly the ...
user6247850's user avatar
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